2 Answers
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The Royal Proclamation of 1763 established the British colonies of Quebec, East Florida, West Florida, and Grenada. Among other things, it introduced English law with the words:

... for which Purpose We have given Power under our Great Seal to the
Governors of our said Colonies respectively to erect and constitute,
with the Advice of our said Councils respectively, Courts of
Judicature and public Justice within our Said Colonies for hearing and
determining all Causes, as well Criminal as Civil, according to Law
and Equity, and as near as may be agreeable to the Laws of England,
with Liberty to all Persons who may think themselves aggrieved by the
Sentences of such Courts, in all Civil Cases, to appeal, under the
usual Limitations and Restrictions, to Us in our Privy Council. ...

The case Campbell v Hall decided in 1774 (actually about Grenada) suggested that the position was in fact unclear, so the Quebec Act of 1774 was enacted, which tolerated Catholicism and restored French civil law to Quebec, but not criminal law. The American Patriots regarded this as one of the Intolerable Acts, leading the the revolution two years later.

British colonial rule after the end of the Seven Year's War ended in 1763 gave the Canadians, including those in Quebec, little choice. Since the British ruled, their laws became the law of the land. It really wasn't so much of a transition as it was an imposition, in that the British imposed it upon the Canadians.

In 1774, however, the British Parliament passed the Quebec Act in an attempt to ensure Canadian loyalty in the even that Britain went to war with the American colonies. This act restored French civil law, but retained English criminal law. In addition, the Catholic church was once again recognized and permitted to operate openly, and the French language was given official approval.

From 1840, Montreal politics were controlled primarily by English Protestants who tended to favor the British Laws. From 1873 to 1914 there was a transition of power that saw more French Canadians taking part in local government. After 1914, there was basically little or no English influence on politics.

Throughout all this time, even though the British ahd greater political influence, the legal structure remained basically the same. The civil law was based on the French Laws, while the criminal law was based on Enbglish Laws.